A general iterative scheme for k-strictly pseudo-contractive mappings and optimization problems

Applied Mathematics and Computation 217 (2011) 5581–5588

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A general iterative scheme for k-strictly pseudo-contractive mappings and optimization problems q Jong Soo Jung Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

a r t i c l e

i n f o

a b s t r a c t In this paper, we introduce a new general iterative scheme for finding fixed points of a strictly pseudo-contractive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a fixed point of the mapping, which is a solution of a certain optimization problem related to a strongly positive bounded linear operator. Additional results of the main result are also obtained.  2010 Elsevier Inc. All rights reserved.

Keywords: k-strictly pseudo-contractive mapping Nonexpansive mapping Fixed points Contraction Optimization problem Strongly positive bounded linear operator

1. Introduction Let H be a real Hilbert space with inner product h  ,  i and induced norm k  k. Let C be a nonempty closed convex subset of H. A mapping T : C ! H is said to be k-strictly pseudo-contractive if there exists a constant k 2 [0, 1) such that

kTx  Tyk2 6 kx  yk2 þ kkðI  TÞx  ðI  TÞyk2 ;

8x; y 2 C;

ð1:1Þ

and we denote the set of fixed points of the mapping T by F(T); that is, F(T) = fx 2 C : Tx ¼ xg. It is clear that, in a real Hilbert space H, (1.1) is equivalent to

hTx  Ty; x  yi 6 kx  yk2 

1k kðI  TÞx  ðI  TÞyk2 ; 2

8x; y 2 C:

The mapping T is pseudo-contractive if and only if

hTx  Ty; x  yi 6 kx  yk2 ;

8x; y 2 C:

T is strongly pseudo-contractive if and only if there exists a positive constant k 2 (0, 1) such that

hTx  Ty; x  yi 6 ð1  kÞkx  yk2 ;

8x; y 2 C:

We note that the class of k-strictly pseudo-contractions includes the class of nonexpansive mappings T on C (i.e., kTx  Tyk 6 kx  yk, "x, y 2 C) as a subclass. That is, T is nonexpansive if and only if T is 0-strictly pseudo-contractive. The mapping T is also said to be pseudo-contractive if k = 1 and T is said to be strongly pseudo-contractive if there exists a positive constant k 2 (0, 1) such that T  kI is pseudo-contractive. Clearly, the class of k-strictly pseudo-contractive mappings falls into the one between classes of nonexpansive mappings and pseudo-contractive mappings. Also we remark that the class of strongly pseudo-contractive mappings is independent of the class of k-strictly pseudo-contractive mappings (see [1–3]). q

This study was supported by research funds Dong-A University. E-mail address: [email protected]

0096-3003/$ - see front matter  2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.12.034

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The class of pseudo-contractions is one of the most important classes of mappings among nonlinear mappings. Recently, many authors have been devoting the studies on the problems of finding fixed points for pseudo-contractions, see, for example, [4–7] and the references therein. In order to study the fixed point problem for a nonexpansive mapping T, one recent way is to construct the iterative scheme, the so-called viscosity iteration method:

xnþ1 ¼ an f ðxn Þ þ ð1  an ÞTxn ;

n P 0;

ð1:2Þ

where an 2 (0, 1) and f is a contraction of C into itself with constant a 2 (0, 1) (i.e., kf(x)  f(y)k 6 akx  yk, " x, y 2 C). This iterative scheme was first introduced by Moudafi [8]. In particular, under the control conditions

ðC1Þ lim an ¼ 0; n!1

ðC4Þ lim

n!1

ðC2Þ

1 X

an ¼ 1; ðC3Þ

n¼0

1 X

janþ1  an j < 1; or;

n¼0

anþ1 ¼ 1; an

Xu [9] proved that the sequence {xn} generated by (1.2) converges strongly to a fixed point q of T, which is the unique solution of the following variational inequality:

hq  f ðqÞ; q  pi 6 0;

p 2 FðTÞ:

On the other hand, the following optimization problem has been studied extensively by many authors:

min x2X

l 2

1 hAx; xi þ kx  uk2  hðxÞ; 2

1 where X ¼ \1 n¼1 C n ; C 1 ; C 2 ; . . . are infinitely many closed convex subsets of H such that \n¼1 C n – ;; u 2 H;

l P 0 is a real num-

ber, A is a strongly positive bounded linear operator on H (i.e., there is a constant c > 0 such that hAx; xi P ckxk2 ; 8x 2 H) and h is a potential function for cf (i.e., h0 (x) = cf(x) for all x 2 H). For this kind of optimization problems, see, for example, Bauschke and Borwein [10], Combettes [11], Deutsch and Yamada [12], Jung [13] and Xu [14] when X ¼ \Ni¼1 C i and h(x) = hx, bi for a given point b in H. In 2007, related to a certain optimization problem, Marino and Xu [15] introduced the following general iterative scheme for the fixed point problem of a nonexpansive mapping:

xnþ1 ¼ an cf ðxn Þ þ ðI  an AÞSxn ;

n P 0;

ð1:3Þ

where {an} 2 (0, 1) and c > 0. They proved that the sequence {xn} generated by (1.3) converges strongly to the unique solution of the variational inequality

hðA  cf Þx ; x  x i P 0;

x 2 FðSÞ;

which is the optimality condition for the minimization problem

min

x2FðSÞ

1 hAx; xi  hðxÞ; 2

where h is a potential function for cf. The result improved the corresponding results of Moudafi [8] and Xu [9]. In 2009, Cho et al. [5] extended the result of Marino of Xu [15] to the class of k-strictly pseudo-contractive mappings as follows: Theorem CKQ. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C  C, and T : C ! H be a k-strictly pseudocontractive mapping with F(T) – ; for some 0 6 k < 1. Let A be a strongly positive bounded linear operator on C with coefficient c and f : C ! C be a contraction with the contractive coefficient 0 < a < 1 such that 0 < c < ac. Let {an}  (0, 1). Let x0 2 C and let {xn} be a sequence in C generated by

xnþ1 ¼ an cf ðxn Þ þ ðI  an AÞP C Sxn ;

n P 0;

where S : C ! H is a mapping defined by Sx = kx + (1  k)Tx and PC is the metric projection of H onto C. Let {an} be satisfy conditions (C1), (C2) and (C3). Then {xn} converges strongly to a fixed point q of T, which is the unique solution of the following variational inequality related to the linear operator A:

hcf ðqÞ  Aq; p  qi 6 0;

p 2 FðTÞ:

In 2010, in order to improve the corresponding results of Cho et al. [5] as well as Marino and Xu [15] by removing the condition (C3) in Theorem CKQ, Jung [6] studies the following composite iterative scheme.

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Theorem J. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C  C, and T : C ! H be a k-strictly pseudocontractive mapping with F(T) – ; for some 0 6 k < 1. Let A be a strongly positive bounded linear operator on C with coefficient c and f : C ! C be a contraction with the contractive coefficient 0 < a < 1 such that 0 < c < ac. Let {an} and {bn} be sequences in (0, 1). Let x0 2 C and let {xn} be a sequence in C generated by



yn ¼ bn xn þ ð1  bn ÞP C Sxn ; xnþ1 ¼ an cf ðxn Þ þ ðI  an AÞyn ;

n P 0;

where S : C ! H is a mapping defined by Sx = kx + (1  k)Tx and PC is the metric projection of H onto C. Let {an} be satisfy the conditions (C1) and (C2) and let {bn} be satisfy the condition 0 < liminfn?1 bn 6limsupn?1bn < 1. Then {xn} converges strongly to a fixed point q of T, which is the unique solution of the following variational inequality related to the linear operator A:

hcf ðqÞ  Aq; p  qi 6 0;

p 2 FðTÞ:

In this paper, motivated by the above-mentioned results, we introduce a new general iterative scheme for finding fixed points of a strictly pseudo-contractive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a fixed point of the mapping, which is a solution of a certain optimization problem related to a strongly positive linear operator. Additional results of the main result are also obtained. Our results improve and develop the corresponding results of Cho et al. [5], Jung [6] and Marino and Xu [15] as well as Halpern [16], Moudafi [8], Wittmann [17] and Xu [9]. 2. Preliminaries and lemmas Let H be a real Hilbert space and C be a nonempty closed convex subset of H. In the following, when {xn} is a sequence in H, xn ? x (resp., xn N x) will denote strong (resp., weak) convergence of the sequence {xn} to x. For every point x 2 H, there exists a unique nearest point in C, denoted by PC(x), such that

kx  PC ðxÞk 6 kx  yk for all y 2 C. PC is called the metric projection of H onto C. It is well known that PC is nonexpansive. It is also well known that H satisfies the Opial condition (cf. [18]), that is, for any sequence {xn} with xn N x, the inequality

lim inf kxn  xk < lim inf kxn  yk n!1

n!1

holds for every y 2 H with y – x. We need the following lemmas for the proof of our main results. Lemma 2.1 [19]. Let H be a Hilbert space and C be a closed convex subset of H. If T is a k-strictly pseudo-contractive mapping on C, then the fixed point set F(T) is closed convex, so that the projection PF(T) is well defined. Lemma 2.2 [19]. Let H be a Hilbert space and C be a closed convex subset of H. Let T : C ! H be a k-strictly pseudo-contractive mapping with F(T) – ;. Then F(PCT) = F(T). Lemma 2.3 [19]. Let H be a Hilbert space, C be a closed convex subset of H, and T : C ! H be a k-strictly pseudo-contractive mapping. Define a mapping S : C ? H by Sx = kx + (1  k)Tx for all x 2 C. Then, as k 2 [k, 1), S is a nonexpansive mapping such that F(S) = F(T). The following Lemmas 2.4 and 2.5 can be obtained from the Proposition 2.6 of Acedo and Xu [4]. Lemma 2.4. Let H be a Hilbert space and C be a closed convex subset of H. For any N P 1, assume that for each 1 6 i 6 N, T i : C ? H is a ki-strictly pseudo-contractive mapping for some 0 6 ki < 1. Assume that fgi gN i¼1 is a positive sequence such that PN PN i¼1 gi ¼ 1. Then i¼1 gi T i is a nonself-k-strictly pseudo-contractive mapping with k = maxfki : 1 6 i 6 Ng. Lemma 2.5. Let fT i gNi¼1 and fgi gNi¼1 be given as in Lemma 2.4. Suppose that fT i gNi¼1 has a common fixed point in C. Then P Fð Ni¼1 gi T i Þ ¼ \Ni¼1 FðT i Þ. Lemma 2.6 [15]. Assume that A is a strongly positive linear bounded operator on a Hilbert space H with coefficient c > 0 and 0 < q 6 kAk1. Then kI  qAk 6 1  qc. Lemma 2.7 ([14,20]). Let {sn} be a sequence of non-negative real numbers satisfying

snþ1 6 ð1  kn Þsn þ kn dn þ r n ;

n P 0;

where {kn}, {dn} and {rn} satisfy the following conditions: P k ¼ 1, (i) {kn}  [0, 1] and 1 n¼0 Pn (ii) limsupn?1dn 6 0 or 1 n¼0 kn dn < 1;

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J.S. Jung / Applied Mathematics and Computation 217 (2011) 5581–5588

(iii) rn P 0 ðn P 0Þ;

P1

n¼0 r n

< 1.

Then limn?1sn = 0 Lemma 2.8 [21]. Let {xn} and {zn} be bounded sequences in a Banach space E and {cn} be a sequence in [0, 1] which satisfies the following condition:

0 < lim inf cn 6 lim sup cn < 1: n!1

n!1

Suppose that xn+1 = cnxn + (1  cn)zn, n P 1, and

lim supðkznþ1  zn k  kxnþ1  xn kÞ 6 0: n!1

Then limn?1kzn  xnk = 0. Lemma 2.9. In a Hilbert space H, the following inequality holds:

kx þ yk2 6 kxk2 þ 2hy; x þ yi;

x; y 2 H:

Finally, the following lemma can be found in [22] (see also Lemma 2.1 in [13]). Lemma 2.10. Let C be a nonempty closed convex subset of a real Hilbert space H, and g : C ! R [ f1g be a proper lower semicontiunous differentiable convex function. If x⁄ is a solution to the minimization problem

gðx Þ ¼ inf gðxÞ; x2C

then

hg 0 ðxÞ; x  x i P 0;

x 2 C:

⁄

In particular, if x solves the optimization problem

min x2C

l 2

1 hAx; xi þ kx  uk2  hðxÞ; 2

then

hu þ ðcf  ðI þ lAÞÞx ; x  x i 6 0;

x 2 C;

where h is a potential function for cf. 3. Main results Now we study the strong convergence of a general iterative scheme. Theorem 3.1. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C  C, and T : C ! H be a k-strictly pseudocontractive mapping with F(T) – ; for some k 2 (0, 1). Let l > 0. Let A be a strongly positive bounded linear operator on C with constant c 2 ð0; 1Þ and f : C ! C be a contraction with the contractive coefficient a 2 (0, 1) such that 0 < c < ð1þalÞc. Let {an} and {bn} be sequences in (0, 1) which satisfy the conditions: (C1) limn?1an = 0; P1 (C2) n¼0 an ¼ 1; (B) 0 < lim infn?1bn 6 lim supn?1bn < 1. Let u 2 C be a fixed element. Let x0 2 C and let {xn} be a sequence in C generated by

xnþ1 ¼ an ðu þ cf ðxn ÞÞ þ bn xn þ ðð1  bn ÞI  an ðI þ lAÞÞPC Sxn ;

n P 0;

ðISÞ

where S : C ! H is a mapping defined by Sx = kx + (1  k)Tx and PC is the metric projection of H onto C. Then {xn} converges strongly to a fixed point of T, which is a solution of the optimization problem

min

x2FðTÞ

l 2

1 hAx; xi þ kx  uk2  hðxÞ; 2

where h is a potential function for cf.

ðOP1Þ

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Proof. First we note that from the condition (C1), we may assume, without loss of generality, that an 6 (1 + lkAk)1 and 2ðð1þlÞccaÞ an < 1 for n P 0. Recall that if A is bounded linear self-adjoint operator on H, then 1an ca

kAk ¼ supfjhAu; uij : u 2 H; kuk ¼ 1g: Observe that

hðð1  bn ÞI  an ðI þ lAÞÞu; ui ¼ 1  bn  an  an lhAu; ui P 1  bn  an  an lkAk P 0; which is to say (1  bn)I  an(I + lA) is positive. It follows that

kð1  bn ÞI  an ðI þ lAÞk ¼ sup fhðð1  bn ÞI  an ðI þ lAÞÞu; ui : u 2 H; kuk ¼ 1g ¼ supf1  bn  an  an lhAu; ui : u 2 H; kuk ¼ 1g 6 1  bn  an ð1 þ lcÞ < 1  bn  an ð1 þ lÞc: ð3:1Þ Now we divide the proof into several steps: Step 1. We show that {xn} is bounded. To this end, let p 2 F(T) and set A ¼ ðI þ lAÞ. Then, from (IS) and (3.1), we obtain

kxnþ1  pk ¼ kan u þ an ðcf ðxn Þ  ApÞ þ bn ðxn  pÞ þ ðð1  bn ÞI  an AÞðP C Sxn  pÞk 6 ð1  bn  ð1 þ lcÞan Þkxn  pk þ bn kxn  pk þ an kuk þ an ckf ðxn Þ  f ðpÞk þ an kcf ðpÞ  Apk 6 ð1  ð1 þ lÞcan Þkxn  pk þ an kuk þ an cakxn  pk þ an kcf ðpÞ  Apk ¼ ð1  ðð1 þ lÞc  caÞan Þkxn  pk þ ðð1 þ lÞc  caÞan

kcf ðpÞ  Apk þ kuk : ð1 þ lÞc  ca

ð3:2Þ

From (3.2), it follows that

(

) kcf ðpÞ  Apk þ kuk : kxnþ1  pk 6 max kxn  pk; ð1 þ lÞc  ca

ð3:3Þ

By induction, it follows from (3.3) that

( kxn  pk 6 max kx0  pk;

) kcf ðpÞ  Apk þ kuk n P 1: ð1 þ lÞc  ca

Therefore {xn} is bounded. So {f(xn)}, fAP C Sxn g and {PCSxn} are also bounded. Step 2. We show that limn?1kxn+1  xnk = 0. To this show, define

xnþ1 ¼ bn xn þ ð1  bn Þzn ;

for all n P 0:

Observe that from the definition of zn, we have

znþ1  zn ¼ ¼ ¼ ¼

xnþ2  bnþ1 xnþ1 xnþ1  bn xn  1  bnþ1 1  bn

anþ1 ðu þ cf ðxnþ1 ÞÞ þ ðð1  bnþ1 ÞI  anþ1 AÞPC Sxnþ1 1  bnþ1

anþ1 1  bnþ1

anþ1 1  bnþ1

ðu þ cf ðxnþ1 ÞÞ 

an 1  bn



an ðu þ cf ðxn ÞÞ þ ðð1  bn ÞI  an AÞPC Sxn 1  bn

ðu þ cf ðxn ÞÞ þ PC Sxnþ1  PC Sxn þ

ðu þ cf ðxnþ1 Þ  AP C Sxnþ1 Þ þ

an 1  bn

an 1  bn

AP C Sxn 

anþ1 1  bnþ1

AP C Sxnþ1

ðAP C Sxn  u  cf ðxn ÞÞ þ PC Sxnþ1  PC Sxn :

Thus, it follows that

kznþ1  zn k  kxnþ1  xn k 6

anþ1 1  bnþ1

ðkuk þ ckf ðxnþ1 Þk þ kAP C Sxnþ1 kÞ þ

From the condition (C1) and (B), it follows that

lim supðkznþ1  zn k  kxnþ1  xn kÞ 6 0: n!1

Hence, by Lemma 2.8, we have

lim kzn  xn k ¼ 0:

n!1

Consequently,

lim kxnþ1  xn k ¼ lim ð1  bn Þkzn  xn k ¼ 0:

n!1

n!1

an 1  bn

ðkAP C Sxn k þ kuk þ ckf ðxn ÞkÞ:

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J.S. Jung / Applied Mathematics and Computation 217 (2011) 5581–5588

Step 3. We show that limn?1kxn  PCSxnk = 0. Indeed, since

xnþ1 ¼ an ðu þ cf ðxn ÞÞ þ bn xn þ ðð1  bn ÞI  an AÞPC Sxn ; we have

kxn  PC Sxn k 6 kxn  xnþ1 k þ kxnþ1  PC Sxn k 6 kxn  xnþ1 k þ an ðkuk þ ckf ðxn Þ  AP C Sxn kÞ þ bn kxn  PC Sxn k; that is,

kxn  PC Sxn k 6

1 an kxn  xnþ1 k þ ku þ cf ðxn Þ  AP C Sxn k: 1  bn 1  bn

So, from the conditions (C1) and (B) and Step 2, it follows that

lim kxn  PC Sxn k ¼ 0:

n!1

Step 4. We show that

lim suphu þ ðcf  ðI þ lAÞÞq; xn  qi ¼ lim suphu þ ðcf  AÞq; xn  qi 6 0; n!1

n!1

where q is a solution of the optimization problem (OP1). To show this, we can choose a subsequence fxnj g of {xn} such that

limhu þ ðcf  AÞq; xnj  qi ¼ lim suphu þ ðcf  AÞq; xn  qi: j!1

n!1

Since {xn} is bounded, there exists a subsequence fxnj g of fxnj g which converges weakly to w. Without loss of generality, we i

can assume that xnj * w. Since kxn  PCSxnk ? 0 by Step 3, we obtain w = PCSw. In fact, if w – PCSw, then, by Opial condition,

lim inf kxnj  wk < lim inf kxnj  PC Swk 6 lim inf ðkxnj  PC Sxnj k þ kPC Sxnj  PC SwkÞ 6 lim inf kxnj  wk; j!1

j!1

j!1

j!1

which is a contradiction. Hence w = PCSw. Since F(PCS) = F(S), from Lemma 2.3, we have w 2 F(T). Therefore, from Lemma 2.10, we conclude that

lim suphu þ ðcf  ðI þ lAÞÞq; xn  qi ¼ limhu þ ðcf  AÞq; xnj  qi ¼ hu þ ðcf  AÞq; w  qi 6 0: j!1

n!1

Step 5. We show that {xn} converges strongly to q 2 F(T), which is a solution of the optimization problem (OP1). Indeed, from (IS) and Lemma 2.9, we have

kxnþ1  qk2 ¼ kan ðu þ cf ðxn Þ  AqÞ þ bn ðxn  qÞ þ ðð1  bn ÞI  an AÞðPC Sxn  qÞk 6 kbn ðxn  qÞ þ ðð1  bn ÞI  an AÞðPC Sxn  qÞk2 þ 2an hu þ cf ðxn Þ  Aq; xnþ1  qi 6 ðkðð1  bn ÞI  an AÞðPC Sxn  qÞk þ kbn ðxn  qÞkÞ2 þ 2an chf ðxn Þ  f ðqÞ; xnþ1  qi þ 2an hu þ cf ðqÞ  Aq; xnþ1  qi 6 ðð1  bn  ð1 þ lÞcan Þkxn  qk þ bn kxn  qkÞ2 þ 2an cakxn  qkkxnþ1  qk þ 2an hu þ cf ðqÞ  Aq; xnþ1  qi 6 ð1  ð1 þ lÞcan Þ2 kxn  qk2 þ 2an cakxn  qkkxnþ1  qk þ 2an hu þ ðcf  AÞq; xnþ1  qi 6 ð1  ð1 þ lÞcÞan Þ2 kxn  qk2 þ an ca½kxn  qk2 þ kxnþ1  qk2  þ 2an hu þ ðcf  AÞq; xnþ1  qi; that is,

1  2ð1 þ lÞcan þ ðð1 þ lÞcÞ2 a2n þ an ca 2an kxn  qk2 þ hu þ ðcf  AÞq; xnþ1  qi 1  an ca 1  an ca   2ðð1 þ lÞc  caÞan ðð1 þ lÞcÞ2 a2n 2an kxn  qk2 þ kxn  qk2 þ hu þ ðcf  AÞq; xnþ1  qi ¼ 1 1  an ca 1  an ca 1  an ca   2ðð1 þ lÞc  caÞ 2ðð1 þ lÞc  caÞan 6 1 an kxn  qk2 þ 1  an ca 1  an ca ! 2 ðð1 þ lÞcÞ an 1  hu þ ðcf  AÞq; xnþ1  qi M1 þ ð1 þ lÞc  ca 2ðð1 þ lÞc  caÞ

kxnþ1  qk2 6

¼ ð1  kn Þkxn  qk2 þ kn dn ;

J.S. Jung / Applied Mathematics and Computation 217 (2011) 5581–5588

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lÞccaÞ where M 1 ¼ supfkxn  qk2 : n P 0g; kn ¼ 2ðð1þ an and 1an ca

dn ¼

ðð1 þ lÞcÞ2 an 1 hu þ ðcf  AÞq; xnþ1  qi: M1 þ ð1 þ lÞc  ca 2ðð1 þ lÞc  caÞ

From the conditions (C1) and (C2) and Step 4, it is easy to see that kn ! 0; Lemma 2.7, we conclude xn ? q as n ? 1. This completes the proof. h

P1

n¼0 kn

¼ 1 and limsupn?1dn 6 0. Hence, by

Remark 3.1. P (1) Theorem 3.1 improves Theorem 2.1 of Cho et al. [5] by removing the condition (C3) 1 n¼0 janþ1  an j < 1 in a general iterative scheme related to an optimization problem. (2) Theorem 3.1 also develops Theorem 2.1 of Jung [6]. (3) Theorem 3.1 includes the corresponding results of Halpern [16], Marino and Xu [15], Wittmann [17] and Xu [9] as some special cases. Theorem 3.2. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C  C, and T i : C ! H be a ki-strictly pseudocontractive mapping for some 0 6 ki < 1 and \Ni¼1 FðT i Þ – ;. Let l > 0. Let A be a strongly positive bounded linear operator on C with constant c 2 ð0; 1Þ and f : C ? C be a contraction with the contractive coefficient 0 < a < 1 such that 0 < c < ð1þalÞc. Let {an} and {bn} be sequences in (0, 1) which satisfy the conditions: (C1) limn?1an = 0; P1 (C2) n¼0 an ¼ 1; (B) 0 < lim infn?1bn 6 lim supn?1bn < 1. Let u 2 C be a fixed element. Let x0 2 C and let {xn} be a sequence in C generated by

xnþ1 ¼ an ðu þ cf ðxn ÞÞ þ bn xn þ ðð1  bn ÞI  an ðI þ lAÞÞPC Sxn ; n P 0; P where S : C ? H is a mapping defined by Sx ¼ kx þ ð1  kÞ Ni¼1 gi T i x with k = maxfki : 1 6 i 6 N} and {gi} is a positive sequence PN such that i¼1 gi ¼ 1 and PC is the metric projection of H onto C. Then {xn} converges strongly to a common fixed point of fT i gNi¼1 , which is a solution of the optimization problem

min

x2\N FðT i Þ i¼1

l 2

1 hAx; xi þ kx  uk2  hðxÞ; 2

ðOP2Þ

where h is a potential function for cf. P Proof. Define a mapping T : C ? H by Tx ¼ Ni¼1 gi T i x. By Lemmas 2.4 and 2.5, we conclude that T : C ? H is a k-strictly P pseudo-contractive mapping with k = maxfki : 1 6 i 6 Ng and FðTÞ ¼ Fð Ni¼1 gi T i Þ ¼ \Ni¼1 FðT i Þ. Then the result follows from Theorem 3.1 immediately. h As a direct consequence of Theorem 3.2, we have the following results for nonexpansive mappings (that is, 0-strictly pseudo-contractive mappings). Theorem 3.3. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C  C, and fT i gN i¼1 : C ! H be a finite family of nonexpansive mappings with \N FðT Þ–;. Let l > 0. Let A be a strongly positive bounded linear operator on C with constant i i¼1

c 2 ð0; 1Þ and f : C ? C be a contraction with the contractive coefficient 0 < a < 1 such that 0 < c < ð1þalÞc. Let {an} and {bn} be sequences in (0, 1) which satisfy the conditions: (C1) limn?1an = 0; P1 (C2) n¼0 an ¼ 1; (B) 0 < lim infn?1bn 6 lim supn?1bn < 1. Let u 2 C be a fixed element. Let x0 2 C and let {xn} be a sequence in C generated by

xnþ1 ¼ an ðu þ cf ðxn ÞÞ þ bn xn þ ðð1  bn ÞI  an ðI þ lAÞÞPC

N X

gi T i xn ; n P 0;

i¼1

P where fgi gNi¼1 is a positive sequence such that Ni¼1 gi ¼ 1 and PC is the metric projection of H onto C. Then {xn} converges strongly to a common fixed point of fT i gNi¼1 , which is a solution of the optimization problem

5588

J.S. Jung / Applied Mathematics and Computation 217 (2011) 5581–5588

min

x2\N FðT i Þ i¼1

l 2

1 hAx; xi þ kx  uk2  hðxÞ; 2

ðOP3Þ

where h is a potential function for cf. Remark 3.2. Theorems 3.2 and 3.3 improve and develop the corresponding results of Cho et al. [5], Jung [6] and Marino and Xu [15]. Acknowledgement The author thanks the referees for their valuable comments and suggestions, which improved the presentation of this manuscript. References [1] F.E. Browder, Fixed point theorems for noncompact mappings, Proc. Natl. Acad. Sci. USA. 53 (1965) 1272–1276. [2] F.E. Browder, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Ration. Mech. Anal. 24 (1967) 82–90. [3] F.E. Browder, W.V. Petryshn, Construction of fixed points of nonlinear mappings Hilbert space, J. Math. Anal. Appl. 20 (1967) 197–228. [4] G.L. Acedo, H.K. Xu, Iterative methods for strictly pseudo-contractions in Hilbert space, Nonlinear Anal. 67 (2007) 2258–2271. [5] Y.J. Cho, S.M. Kang, X. Qin, Some results on k-strictly pseudo-contractive mappings in Hilbert spaces, Nonlinear Anal. 70 (2009) 1956–1964. [6] J.S. Jung, Strong convergence of iterative methods for k-strictly pseudo-contractive mappings in Hilbert spaces, Appl. Math. Comput. 215 (2010) 3746– 3753. [7] C.H. Morales, J.S. Jung, Convergence of paths for pseudo-contractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000) 3411–3419. [8] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55. [9] H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291. [10] H.H. Bauschke, J.M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev. 38 (1997) 367–426. [11] P.L. Combettes, Hilbertian convex feasibility problem: convergence of projection methods, Appl. Math. Optim. 35 (1997) 311–330. [12] F. Deutsch, I. Yamada, Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings, Numer. Funct. Anal. Optim. 19 (1998) 33–56. [13] J.S. Jung, Iterative algorithms with some control conditions for quadratic optimizations, Panamer. Math. J. 16 (4) (2006) 13–25. [14] H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002) 240–256. [15] G. Marino, H.X. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52. [16] B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961. [17] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491. [18] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge Univ. Press, Cambridge, UK, 1990. [19] H. Zhou, Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 69 (2008) 456–462. [20] L.S. Liu, Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114–125. [21] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integral, J. Math. Anal. Appl. 305 (2005) 227–239. [22] Y.H. Yao, M. Aslam Noor, S. Zainab, Y.-C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl. 354 (2009) 319–329.